Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.5

Time: 14.3s

Precision: binary64

Cost: 84224

Math

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\pi \cdot 0.5}\\ \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\mathsf{fma}\left({t_0}^{2}, t_0, -\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0 (cbrt (* PI 0.5))))
   (expm1
    (pow
     (sqrt
      (log1p
       (fma
        (pow t_0 2.0)
        t_0
        (- (asin (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))))
     2.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

↓

double code(double v) {
	double t_0 = cbrt((((double) M_PI) * 0.5));
	return expm1(pow(sqrt(log1p(fma(pow(t_0, 2.0), t_0, -asin((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)))))), 2.0));
}

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

↓

function code(v)
	t_0 = cbrt(Float64(pi * 0.5))
	return expm1((sqrt(log1p(fma((t_0 ^ 2.0), t_0, Float64(-asin(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))))) ^ 2.0))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[v_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision]}, N[(Exp[N[Power[N[Sqrt[N[Log[1 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + (-N[ArcSin[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Initial program 0.5

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

3. Applied egg-rr 0.5

   \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{2}}\right) \]

4. Applied egg-rr 0.5

   \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}^{2}\right) \]

5. Final simplification 0.5

   \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\mathsf{fma}\left({\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}\right)}^{2}\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	45440

\[\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{2}\right) \]

Alternative 2
Error	0.5
Cost	7232

\[\cos^{-1} \left(\frac{1 + -5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right) \]

Alternative 3
Error	1.3
Cost	6464

\[\cos^{-1} -1 \]

Reproduce

herbie shell --seed 2022312 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))